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If y=tan^(-1)(1/(1+x+x^2))+tan^(-1)(1/(x...

If `y=tan^(-1)(1/(1+x+x^2))+tan^(-1)(1/(x^2+3x+3))+tan^(-1)(1/(x^2+5x+7))+...+` upto `n` terms, then find the value of `y^(prime)(0)`

Text Solution

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`y=tan^(-1)""(1)/(1+x+x^(2))+tan^(-1)""(1)/(x^(2)+3x+3)+...+"n terms"`
`y=tan^(-1)""((x+1)-x)/(1+x(1+x))+tan^(-1)""((x+2)-(x+1))/(1+(x+1)(x+2))+...+"n terms"`
`=tan^(-1)(x+1)-tan^(-1)x+tan^(-1)(x+2)-tan^(-1)(x+1)+tan^(-1)(x+n)-tan^(-1)(x+(n-1)`
`therefore" "y'(x)=(1)/(1+(x+n)^(2))-(1)/(1+x^(2))`
`or" "y'(0)=(1)/(1+n^(2))-1(-n^(2))/(1+n^(2))`
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